Find the general solution for \( \theta \) if \[ 3 \sin \theta=2 \cos \theta \]

To solve the equation \( 3 \sin \theta = 2 \cos \theta \), we first manipulate it into a more useful form by dividing both sides by \( \cos \theta \) (assuming \( \cos \theta \neq 0 \)): \[ 3 \tan \theta = 2 \] Now, we isolate \( \tan \theta \): \[ \tan \theta = \frac{2}{3} \] The general solution for \( \theta \) when \( \tan \theta = k \) is given by: \[ \theta = \tan^{-1}(k) + n\pi \] where \( n \) is any integer. In this case, we have: \[ \theta = \tan^{-1}\left(\frac{2}{3}\right) + n\pi \] Thus, the general solution for \( \theta \) is: \[ \theta = \tan^{-1}\left(\frac{2}{3}\right) + n\pi \quad (n \in \mathbb{Z}) \] Now, for some extra fun facts: The arctangent function, or \( \tan^{-1} \), can be represented graphically, which shows that it’s a function with a range from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \). The periodicity of the tangent function means that once you find one solution, you can easily generate all others by adding or subtracting multiples of \( \pi \). If you decide to calculate \( \tan^{-1}(2/3) \) using a calculator, you'll find that it’s approximately \( 0.588\) radians (or about \( 33.69\) degrees). This comes in handy in variousreal-world applications, especially in engineering and physics, where understanding the angles and relationships of forces are crucial!